Friday, December 07, 2007

Now this is the declaration of our Faith, that we say that God is One, neither dividing His Son from Him, as do the heathen, nor denying, with the Jews, that He was begotten of the Father before all worlds, and afterwards born of the Virgin; nor yet, like Sabellius, confounding the Father with the Word, and so maintaining that Father and Son are one and the same Person; nor again, as doth Photinus, holding that the Son first came into existence in the Virgin’s womb: nor believing, with Arius, in a number of diverse Powers, and so, like the benighted heathen, making out more than one God. For it is written: "Hear, O Israel: the Lord thy God is one God."

Tuesday, December 04, 2007

We, therefore, both know and confess that God is without beginning, without end, eternal and everlasting, uncreate, unchangeable, invariable, simple, uncompound, incorporeal, invisible, impalpable, uncircumscribed, infinite, incognisable, indefinable, incomprehensible, good, just, maker of all things created, almighty, all-ruling, all-surveying, of all overseer, sovereign, judge; and that God is One, that is to say, one essences; and that He is known, and has His being in three subsistences, in Father, I say, and Son and Holy Spirit; and that the Father and the Son and the Holy Spirit are one in all respects, except in that of not being begotten, that of being begotten, and that of procession; and that the Only-begotten Son and Word of God and God, in His bowels of mercy, for our salvation, by the good pleasure of God and the co-operation of the Holy Spirit, being conceived without seed, was born uncorruptedly of the Holy Virgin and Mother of God, Mary, by the Holy Spirit, and became of her perfect Man; and that the Same is at once perfect God and perfect Man, of two natures, Godhead and Manhood, and in two natures possessing intelligence, will and energy, and freedom, and, in a word, perfect according to the measure and proportion proper to each, at once to the divinity, I say, and to the humanity, yet to one composite persons; and that He suffered hunger and thirst and weariness, and was crucified, and for three days submitted to the experience of death and burial, and ascended to heaven, from which also He came to us, and shall come again. And the Holy Scripture is witness to this and the whole choir of the Saints.

And so I say to you, if you wish to become men of might and effectiveness,--"Lift up your minds." However humble any of us may be, however ordinary and common-place in mind, our nature is prefigured after the universal; all the cardinal facts of the universe are the common heritage of humanity; all the prime ideas which pertain to the Godhead and to humanity, all the ground ideas and principles which abide in the realms of mind and spirit!

I have previously noted that it is possible, with a few suppositions, to do propositional logic with Lewis Carroll's Game of Logic, which uses a board consisting of a literal diagram. Once propositional connectives are defined, each sentence in propositional logic can have a corresponding configuration of the board. The Game of Logic understood in this way is truth-functional; all the truth-functional relations and equivalences between propositions are expressible in the relations between configurations of the board, because every configuration of the board is logically isomorphic to a truth table. Thus this is the biliteral configuration of (P & Q):

&nbsp

x

x

x

This is the biliteral configuration of (P v Q):

x

And so forth; we can see immediately, for instance, that (P v Q) is weaker than (P & Q), and how.

But this is the Game of Logic, and not only can we have configurations of the board, we can move from configuration to configuration. There are three types of operations that allow us to move from configuration to configuration: information-adding, information-preserving, and information-subtracting. We can, for instance, use an information-subtracting operation to move from:

&nbsp

x

x

to the (P v Q) configuration noted above. When you look closely at these operations, you begin to realize that they are rather familiar; and this is because they are applications of inference rules. When we realize this we find some obvious patterns. Assumption of premises are information-adding operations; elimination rules are all information-subtracting. The standard introduction rules are all information-preserving, with the exception of addition, which we usually take to be disjunction introduction. If we used Tom's suggested candidate for disjunction introduction, this anomaly would be eliminated, and the elegance of the Game would be greatly increased.

What has happened, though, is that the rules of inference, on the one hand, and the truth-functional relations between sentences in the system, on the other hand, have come apart; while the Game is truth-functional, the rules and the sentences are represented by different aspects of it. The truth-functional relations between sentences are found in the relations between possible configurations of the board; the rules of inference are found in the movements from configuration to configuration. On seeing this, we can start playing with the rules of inference. The possible configurations of the board remain what they are; the truth-functional definitions do not change. But movement from configuration to configuration can be modified. The roads remain the same; only the traffic is diverted.

The same can be said of the propositional logic that the Game is here modeling; implication, for instance, is not inference, or vice versa, as the Tortoise taught us, so the fact that a sentence is in a certain truth-functional relation to other sentences doesn't make it a rule of inference. As a matter of fact, for most purposes we take derivable sentences in propositional logic to correspond to rules of inference. The reasons for doing this are rather obvious: it allows us to have derived rules of inference, which can immensely simplify things. But our doing this is not intrinsic to the truth-functional relations among sentences, but the result of a rule-forming rule that allows us to convert sentences into rules of inference. This is so useful that we sometimes don't make much of a distinction. Thus, for instance, we sometimes use 'derivation' to indicate a truth-functional relation among sentences, in which case sentences no one has ever or will ever derive are 'derived', and we sometimes use it to indicate actual derivation. In other words, we sometimes mean a truth-functional relation between configurations, and we sometimes mean actual operations on one configuration to make it other configurations.

Thus the truth-functional relations among configurations do not establish our rules of inference; they merely indicate that there is no truth-functional impediment to them, when we are interested in truth functions (which we virtually always are). That A implies (A v B) may be in the relevant truth-functional relation to other sentences in the system, but that does not follow that we have to reason, "A, therefore A or B." We can simply deny its usefulness as a rule of inference, derived or otherwise, if we have reason to do so; we can accept it but only under restrictions (e.g., that it not be immediately followed by certain sorts of eliminations); we can accept it wholeheartedly. There is no reason, further, why we can't (as Russell has for the Principia, if I recall correctly) have a rule of inference that has no exactly corresponding sentence in the system (although perhaps a particular instance of it, e.g., modus ponens, might); likewise, there is no reason why we have to accept as a rule of inference each one that corresponds to an implication in the sentence-system. There is strictly speaking no truth-functional reason why we have to have only inference rules that pay attention to truth functions at all. Trying to represent tonk in the Game makes for strange and rather pointless play, but play you can (since tonk is simply defined by rules of inference, it's not necessary to change the way the board can be configured in order to introduce it, but only how you move from configuration to configuration).* We can do these things systematically and consistently; we aren't confined merely to allowing or disallowing rules of inference for particular proofs unless we want to be so confined. And through all of this the truth-functional definitions of the connectives remain exactly what they were, the whole truth-functional system remains exactly what it was. All that changes is the rules of inference, and these can be whatever we have reason to make them. It's a Humpty-Dumpty world.

Of course, from this it doesn't follow that we have good reasons for making them this way rather than that. There are very good reasons, reasons that fit very many circumstances, for making the assumptions we normally make.

-----* In a sense Prior's tonk argument at least suggests the other side of the point I'm making here. We might say that regard for truth-functional relations without regard for rules of inference is useless; while regard for rules of inference without regard for truth-functional relations is pointless. This does not require a conflation; relations among sentences are not inferences, and inferences are not relations among sentences. Of course, we may deliberately allow movement in one or both directions. But you need both the possible configurations of the board and ways to move among them; and they are not the same. (In actuality, Prior's tonk argument goes farther than this and holds that even these together are not perfectly adequate for telling us what we are doing in logic, and I would agree with Prior; but that is not relevant here.)

During those days Mary set out and traveled to the hill country in haste to a town of Judah, where she entered the house of Zechariah and greeted Elizabeth. When Elizabeth heard Mary's greeting, the infant leaped in her womb, and Elizabeth, filled with the holy Spirit, cried out in a loud voice and said, "Most blessed are you among women, and blessed is the fruit of your womb. And how does this happen to me, that the mother of my Lord should come to me? For at the moment the sound of your greeting reached my ears, the infant in my womb leaped for joy. Blessed are you who believed that what was spoken to you by the Lord would be fulfilled."

Sunday, December 02, 2007

Reading Todhunter on Whewell (1876), I came across the interesting phrase, "Newton unravelled the texture of solar light," which strikes the ear nicely. Todhunter doesn't explicitly tell us who is the source, but simply says it is "a favourite expression of one of the most eminent of living scientific lectures" (p. 330). I find that the expression is found in John Tyndall's lectures on radiation (1865); a similar expression is also found in his lectures on light (delivered 1872-1873). Todhunter himself borrows the expression in the section on light in his book on natural philosophy (1877). The same phrase applied to Newton's work with light is used in a review of books on the life of Newton that ultimately comes from the Edinburgh Review (1844); this review, as far as I am aware, is anonymous.

Caveats

For a rough introduction to my philosophy of blogging, including the Code of Amiability I try to follow on this weblog, please read my fifth anniversary post. I consider blogging to be a very informal type of publishing - like putting up thoughts on your door with a note asking for comments. Nothing in this weblog is done rigorously: it's a forum to let my mind be unruly, a place for jottings and first impressions. Because I consider posts here to be 'literary seedings' rather than finished products, nothing here should be taken as if it were anything more than an attempt to rough out some basic thoughts on various issues. Learning to look at any topic philosophically requires, I think, jumping right in, even knowing that you might be making a fool of yourelf; so that's what I do. My primary interest in most topics is the flow and structure of reasoning they involve rather than their actual conclusions, so most of my posts are about that. If, however, you find me making a clear factual error, let me know; blogging is a great way to get rid of misconceptions.